A generalization of a theorem of M. Riesz to the case of functions of several variables

The following theorem was proved by M. Riesz: Iff(x) L(–,),f(x) 0 and the conjugate functionf (x) is also integrable on [-, ], thenf(x) L log⁺L. The analog of this theorem for functions of several variables is established.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 269–280, November, 1968.     

Primes in short intervals

Let (x) stand for the number of primes not exceedingx. In the present work it is shown that if 23/421,yx andx>x() then (x)–(x–y)>y/(100 logx). This implies for the difference between consecutive primes the inequalityp _n+1–p _n p _n ^23/42 .     

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, , ) we choose an additive measure: Z (Z is a Banach space) with the following property: for alle , we have ; this measure defines an indefinite integral over the measure onL ² (T, ,). We prove that if { _n (t)} _n ^=1/ is an orthonormal basis inL ² and _n (e)=e _n (t) d, then any additive measure: Z whose Radon-Nikodým derivatived/d belongs toL ² is uniquely expandable in a series(e)= _n ^=1/ _n n(e) that converges to(e) uniformly with respect toe can be differentiated term-by-term, and satisfies _n ^=1/ _n ^/2 <. In the caseL ²[0,2],Z=, the Fourier series of a 2-periodic absolutely continuous functionF(t) such thatF'(t) L ²[0, 2] is superuniformly convergent toF(t).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 180–184, August, 1998.     

The Exterior Cube L-Function for GL(6)

We construct a Rankin Selberg integral to represent the exterior cube L function L(,³,s) of an automorphic cuspidal module of GL₆( _F ) (where F is a number field). We determine the poles of this L function and find period conditions for the special value L(,³,1/2). We use the Siegal Weil formula. We also state an analogue of the Gross–Prasad conjecture concerning a criterion for the nonvanishing of L(,³,1/2).     

Isolated points in duals of certain locally compact groups

Summary LetG be a separable locally compact group with dual space. consists of all equivalence classes of irreducible unitary representations ofG, and is endowed with the Fell-topology. We study the topological properties in of the square-integrable representations ofG. [ is square-integrable provided there is a coordinate functiong((g)v, v),gG, for which is inL ²(G) w.r.t. left Haar measure onG.]SupposeG contains an open normal subgroupN of the formeKN ⁿ e whereK is compact. (All groups with a compact invariant neighborhood of the identity, [IN] groups, satisfy this condition.) In this case we show that if is square-integrable then {} is an open point of.Finally, our techniques are used to prove this result for arbitrary (non connected) nilpotent Lie groups.     

Über nullstellenfreie Gebiete von ζ(itk) (s)

Let ^(itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |^(itk)(s)|, |^(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |^(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt.     

Path properties of an infinite system of Wiener processes

Let be a Poisson process on ^d of intensity and letW ₁(t),W ₂ (t),..., be a sequence of independent Wiener processes. LetW _i (t)=X _i +W _i (t) whereX ₁,X ₂,..., are the points of . Consider the processess(t)=#{i:X _i (t)1}. These and related processes are studied.     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).     

Proof of two conjectures of H. Brezis and L.A. Peletier

In this paper, we consider the problem: –u=N(N–2)u ^p– , u>0 on ; u=0 on , where is a smooth and bounded domain inR ^N, N3, p= , and >0. We prove a conjecture of H. Brezis and L.A. Peletier about the asymptotic behaviour of solutions of this problem which are minimizing for the Sobolev inequality as goes to zero. We give similar results concerning the related problem: –u=N(N–2)u^p+u, u>0 on ; u=0 on , for N is larger than 4.     

Equivalent Characterization of Entire Functions of Exponential Type

In this paper, we show that a necessary and sufficient condition for the fulfillment of the relation s _m ^(k) (f) – f^(k) _p 0 as m , 1 < p < , k 0,1,2,..., is that f B _,p , where B _,p = B L _p (R), and B denotes the subset of all entire functions of exponential type which are bounded on R, B _,p is usually called Paley-Wiener class, and s _m (f) is the unique cardinal spline of degree m – 1 interpolating f at the integers. Moreover, we obtain three equivalent forms for the characterization of the class B _,p .     

Embedding the Linear Structure of Planar Spaces into Projective Spaces

We show that every non-degenerate planar space with v points and planes can be embedded as a linear space into PG(3,q) for some prime power q provided that 1000( - v) v^5/6;     

Mappings that preserve cones in Lobachevskii space

Let ⁿ be n-dimensional Lobachevskii space, and {l_x:x ⁿ} be a family of lines, parallel to a linel ₀, 0ⁿ (in a given direction). Let {c_x:Xⁿ} be a family of circular cones in ⁿ of opening with axes l_X and vertex X. Then, iff:ⁿⁿ(n>2) is a bijective mapping andf(C_x)=C _f(x), it follows thatf is a motion in the space ⁿ.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 687–694, May, 1973.     

The dissection of a polygon into nearly equilateral triangles

Every polygon can be dissected into acute triangles. In this paper we prove that every polygon, such that the interior angles are at least /5, can be dissected into triangles with interior angles all less than or equal to 2/5. We find necessary conditions on the interior angles of the polygon in order to obtain a dissection into triangles with interior angles all (where /3<<2/5). The conjecture can be stated that these conditions are also sufficient.     

On flags in a topological projective plane

In this paper we study some properties of the flag space () of a topological projective plane . The line set, sayL, of (), splits into two subfamiliesL ₀ andL ₁ which an non-trivial partitions of the point set of (). For any two distinct lines , ' belonging to one of the two subfamilies, there is exactly one line in the other subfamily meeting botl and '; turns out to be a continuous mapping of (-,'–. Further, the intersection o: two lines, having exactly one point in common, is a continuous and open mapping. Conversely, every irreducible semilinear space, satisfying the above properties, is isomor phic to the flag space of a topological projective plane.This paper has been prepared from some results of a research project supported by N.A.T.O.     

Приближение функции ограниченнойp-флукту адии полиномами по мульти пликативным система м

A new discrete modulus of continuity is introduced for functions of boundedp-fluctuation, and direct and converse theorems are proved on the approximation of these functions by polynomials with respect to multiplicative systems. Sufficient conditions for the convergence of Fourier series with respect to multiplicative systems are also obtained and these are the best possible in a certain sense.

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Best approximation of periodic functions by trigonometric polynomials in L2

Estimates are gotten for the best approximations in L₂ (0, 2) of a periodic function by trigonometric polynomials in terms of its m-th continuity modulus or in terms of the continuity modulus of its r-th derivative. The inequality E_n–1(f)L₂<(C _2m ^m _m(2/n;f)L₂(1 const) is proved, where the constant (C _2m ^m )^–1/2 is unimprovable for the whole space L₂ (0, 2). Two titles are cited in the bibliography.Translated from Matematicheskie Zametki, Vol. 2, No. 5, pp. 513–522, November, 1967.     

Bildgarben und Fasercohomologie für Relativ Analytische Räume

Let : XY be a relative analytic space and an _X module. The object of this paper is to collect some results on the connection between the fibre cohomology groups Hⁱ(X(y),(y)) and the direct image sheaves Rⁱ _*. This generalizes results of Kodaira-Spencer and Grauert-Riemen-schneider giving a unified approach based on the methods of [9].     

“Quasiphotons” in an active medium

We consider the propagation of concentrated wave packets along space-time rays for the case of frequency-dependent velocity: c=c_o(x,t)+c₁(x,t,), 1. The complex ray method is applied to construct formal asymptotic expansions. It is shown that these packets may propagate only at certain medium-determined frequencies.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 104–115, 1985.In conclusion, I would like to thank I. A. Molotkov for proposing the problem and for useful comments and A. P. Kachalov for fruitful discussion.     

Asymptotic Formulae and Divisor Problems

We investigate the asymptotic behaviour of the summatory functions of _z(n, ), _k(n, ) z ⁽ⁿ⁾ and _k(n, ) z ⁽ⁿ⁾.     

Sturmian Words and the Permutation that Orders Fractional Parts

A Sturmian word is a map W : {0,1} for which the set of {0, 1}-vectors F _n(W) {(W(i), W(i + 1),...,W(i + n – 1)) ^T : i } has cardinality exactly n + 1 for each positive integer n. Our main result is that the volume of the simplex whose n + 1 vertices are the n + 1 points in F _n(W) does not depend on W. Our proof of this motivates studying algebraic properties of the permutation _,n (where is any irrational and n is any positive integer) that orders the fractional parts {}, {2},...,{n}, i.e., 0 < {_,n (1)} < {_,n (2)} < ··· < {_,n (n)} < 1. We give a formula for the sign of _,n , and prove that for every irrational there are infinitely many n such that the order of _,n (as an element of the symmetric group S _n) is less than n.